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Ratios and Rates

How to mix perfect lemonade, spot the better buy, and measure the whole Earth.

A ratio compares two amounts. A rate compares two amounts measured in different units, like miles and hours or dollars and items. You use them every time you follow a recipe, check a price tag, or read a speed limit.

This page shows how to write ratios, build ratio tables, find unit rates, and solve proportions, and why each method works. Then try the practice problems and the Rate Race game.

Jump to practicePlay Rate Race
A tape diagram for lemonade: 2 equal boxes of lemon mix and 3 equal boxes of water, labeled 2 to 3 LemonWater2 : 3

What is a ratio?

A ratio compares two quantities. If a lemonade recipe uses 2 scoops of mix for every 3 cups of water, the ratio of mix to water is 2 to 3.

There are three ways to write the same ratio:

  • 2 : 3 (with a colon)
  • 2 to 3 (in words)
  • 2/3 (as a fraction)

Order matters. Mix to water is 2 : 3, but water to mix is 3 : 2. Always say which quantity comes first.

Did you know?

The word “ratio” comes from Latin, where it meant “reckoning” or “calculation.” It’s related to the word “reason.”

Part-to-part and part-to-whole

Say a bag holds 3 red marbles and 5 green marbles, 8 in all.

  • Part-to-part compares one part with another part: red to green is 3 : 5.
  • Part-to-whole compares one part with the total: red to all marbles is 3 : 8. That means 3/8 of the bag is red.

A part-to-whole ratio is the same thing as a fraction of the whole. A part-to-part ratio is not: 3 : 5 does not mean 3/5 of the marbles are red.

Tape diagram: red beads take 2 equal parts and blue beads take 3 equal parts, 5 parts in all. 30 beads total, so each part is 6 beads: 12 red and 18 blue 66666 RedBlue= 18 blue= 12 red5 parts = 30 beads, so 1 part = 6
A tape diagram for red : blue = 2 : 3 with 30 beads in all.

Example 1: Using a tape diagram

The ratio of red to blue beads is 2 : 3, and there are 30 beads. How many of each?

  1. Draw 2 boxes for red and 3 for blue: 5 equal parts.
  2. 30 ÷ 5 = 6 beads in each part.
  3. Red = 2 × 6 = 12, blue = 3 × 6 = 18. Check: 12 + 18 = 30. ✓

Equivalent ratios and ratio tables

Two ratios are equivalent if they describe the same mix. 2 : 3 and 4 : 6 taste the same: you just made a double batch. To get an equivalent ratio, multiply (or divide) both numbers by the same number.

A ratio table lists equivalent ratios in columns. Every column is the first column times some number.

Ratio table for lemonade: scoops of mix 2, 4, 6, 10 match cups of water 3, 6, 9, 15. Labels show times 2, times 3 and times 5 ScoopsCups water 2461036915 × 2× 3× 5 Every column is the same recipe, just bigger.
A ratio table for 2 : 3. Multiply both rows by the same number.

Example 2: Fill in the table

For 2 : 3, how much water goes with 10 scoops? 10 = 2 × 5, so water = 3 × 5 = 15 cups. How many scoops go with 9 cups? 9 = 3 × 3, so scoops = 2 × 3 = 6.

Example 3: Simplest form

A class has 12 girls and 18 boys. The GCF of 12 and 18 is 6, so girls : boys = 12 : 18 = 2 : 3. Girls to the whole class is 12 : 30 = 2 : 5.

Rates and unit rates

A rate is a ratio of two different kinds of quantities, like 120 miles in 2 hours or $3.60 for 4 pens.

A unit rate tells you how much for one unit: miles in 1 hour, or dollars for 1 pen. To find it, divide by the second quantity. The word “per” means “for each.”

Double number line: 0, 60, 120 and 180 miles line up with 0, 1, 2 and 3 hours, so the car goes 60 miles per hour 060120180 0123 mileshours
A double number line for 120 miles in 2 hours. Halve both: 60 miles in 1 hour.

Example 4: Miles per hour

A car goes 120 miles in 2 hours. 120 ÷ 2 = 60 miles per hour. At that rate, 3 hours takes it 60 × 3 = 180 miles.

Example 5: Unit price

4 pens cost $3.60. $3.60 ÷ 4 = $0.90 per pen. So 7 pens would cost 7 × $0.90 = $6.30.

Comparing unit rates: the better buy

Bigger packs aren’t always cheaper. To compare, find the unit price (price for one item, or one ounce) of each choice. The lower unit price is the better buy.

Example 6: Which pack?

PackPriceUnit price
6 granola bars$4.20$4.20 ÷ 6 = $0.70
10 granola bars$6.50$6.50 ÷ 10 = $0.65

$0.65 < $0.70, so the 10-pack is the better buy, saving 5¢ a bar. (It’s only a good deal if you’ll actually eat 10!)

Proportions and cross products

A proportion is an equation saying two ratios are equal, like 2/3 = 4/6. When one number is missing, you can find it two ways.

  • Scale factor: in 3/4 = x/20, the denominator went from 4 to 20, which is × 5. So x = 3 × 5 = 15.
  • Cross products: in a proportion a/b = c/d, the cross products are equal: a × d = b × c. This works even when the scale factor isn’t a whole number.

Example 7: Cross products

Solve 6/8 = x/20.

  1. Cross products: 8 × x = 6 × 20 = 120.
  2. Divide both sides by 8: x = 120 ÷ 8 = 15.
  3. Check: 6/8 = 3/4 and 15/20 = 3/4. ✓

Example 8: Map scale

On a map, 2 cm stands for 5 km. Two towns are 7 cm apart. Set up 2/5 = 7/x. Cross products: 2 × x = 5 × 7 = 35, so x = 35 ÷ 2 = 17.5 km.

How do we know these methods work?

Why equivalent ratios work

Doubling a recipe means using 2 times as much of everything. For each 2 scoops there are still 3 cups of water, so the taste doesn’t change. Multiplying only one number would change the mix.

Why unit rates let you compare

$4.20 for 6 and $6.50 for 10 are hard to compare because the amounts are different. Dividing puts both on the same footing: price for exactly 1. Now you’re comparing like with like.

Why cross products work

Start with a/b = c/d. Multiply both sides by b × d (doing the same thing to both sides keeps them equal). The left side becomes a × d and the right side becomes c × b. So a × d = b × c.

Check with 2/3 = 4/6: 2 × 6 = 12 and 3 × 4 = 12. ✓ Try 2/3 and 3/4: 2 × 4 = 8 but 3 × 3 = 9, so they’re not equal.

History: measuring the Earth with a proportion

Around 240 BCE, Eratosthenes of Cyrene was the head librarian at the great Library of Alexandria in Egypt. He used a proportion to estimate the size of the whole Earth.

He knew that at noon on midsummer’s day, the Sun was straight overhead at Syene (today’s Aswan), but in Alexandria, to the north, a shadow showed the Sun was about 7.2° from overhead. Assuming the Sun’s rays are parallel, that angle is the slice of the Earth’s circle between the two cities.

According to a later account by the writer Cleomedes, he took the distance between the cities as 5,000 stadia. The proportion is:

7.2° / 360° = 5,000 stadia / C

Since 360 ÷ 7.2 = 50, the distance is 1/50 of the way around, so C = 50 × 5,000 = 250,000 stadia. We don’t know exactly how long his stadion was, but by many estimates his answer was close to the real distance around the Earth (about 40,000 km).

The Earth as a circle with Syene and Alexandria marked on its rim. Radii from the center to each city make a 7.2 degree angle, the highlighted arc between the cities is 5,000 stadia, and parallel vertical sun rays shine straight down a well at Syene but cast a 7.2 degree shadow from a stick at Alexandria Syene7.2° Alexandriared arc = 5,000 stadia↓ Sun’s rays (parallel)7.2° = 1/50 of 360°C = 50 × 5,000= 250,000 stadia
Eratosthenes’ idea (angle and sizes exaggerated, not to scale): the sun shines straight down a well at Syene, while a stick at Alexandria casts a shadow. Because the rays are parallel, the shadow angle equals the 7.2° angle between the cities at Earth’s center.

Practice problems

Try each one on paper first, then open the answer to check your work.

1. A team has 12 girls and 15 boys. Write girls : boys and girls : whole team in simplest form.

12 : 15 = 4 : 5 (divide by 3). There are 27 players, so 12 : 27 = 4 : 9.

2. Complete the ratio table for 3 : 5. 3 : 5 = 6 : ? = 9 : ? = ? : 25

× 2: 6 : 10. × 3: 9 : 15. 25 = 5 × 5, so 15 : 25.

3. 6 muffins cost $4.80. What is the price per muffin?

$4.80 ÷ 6 = $0.80 per muffin.

4. A train travels 150 miles in 2.5 hours. What is its speed in miles per hour?

150 ÷ 2.5 = 60 miles per hour. (Check: 60 × 2.5 = 150.)

5. Better buy: 12 oz of cereal for $3.00, or 20 oz for $4.40?

$3.00 ÷ 12 = $0.25 per oz. $4.40 ÷ 20 = $0.22 per oz. The 20 oz box is the better buy.

6. Solve 4/7 = x/21.

Scale factor 3 (7 × 3 = 21), so x = 4 × 3 = 12. Or cross products: 7x = 84, x = 12.

7. Solve 5/x = 15/24.

Cross products: 15 × x = 5 × 24 = 120, so x = 120 ÷ 15 = 8. Check: 5/8 = 15/24. ✓

8. A recipe uses 3 cups of flour for every 2 cups of sugar. How much sugar goes with 9 cups of flour?

3/2 = 9/s. Cross products: 3s = 18, so s = 6 cups of sugar. (9 is 3 × 3, so 2 × 3 = 6.)

Game: Rate Race

Ten mixed problems: equivalent ratios, unit prices, speeds, better buys, proportions and sharing. Type a number like 12 or 0.75, or A/B for better buys, and press Enter. One point per correct answer. Your best score is saved only on this device. About 3 minutes.

Talk About It

  • Is the biggest package always the better buy? When might the smaller one be the smarter choice?
  • Eratosthenes never left Egypt, but he estimated the size of the whole planet. What did he have to assume, and what could go wrong?
  • Where have you seen the word “per” this week? What rate was it describing?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. Prices and recipes in the examples are made up for practice.